I consider the homogeneous Heat equation in $\textbf{R}^d \times (0,T]$ for a fixed $T>0$ : \begin{cases} \partial_t f - \Delta f = 0 & \text{in } \textbf{R}^d \times (0,T] \\ f=f_0 & \text{on } \textbf{R}^d \times \{t=0\} \end{cases} I want to show the conservation of the $L^1$ norm assuming that $f_0 \geq 0$ (and thus $f \geq 0$ by the maximum principle). I know that formally $$0\stackrel{(1)}{=}\int_{\textbf{R}^d}{\Delta f}=\int_{\textbf{R}^d}{\partial_t f}\stackrel{(2)}{=}\partial_t \left(\int_{\textbf{R}^d}{f}\right)=\partial_t( \| f \|_{L^1})$$ but I got difficulties to justify $(1)$ and $(2)$ without making strong assumptions on $f$. I should also precise that I do not want to use the explicit solution given by the Fourier transform's method.
Can anyone help me ? Many thanks !